A class of hyper-bent functions and Kloosterman sums

Abstract

This paper is devoted to the characterization of hyper-bent functions. Several classes of hyper-bent functions have been studied, such as Charpin and Gong’s family $\sum \limits _{r\in R}\text {Tr}_{1}^{n} (a_{r}x^{r(2^{m}-1)})$ and Mesnager’s family $\sum \limits _{r\in R}\text {Tr}_{1}^{n}(a_{r}x^{r(2^{m}-1)}) +\text {Tr}_{1}^{2}(bx^{\frac {2^{n}-1}{3}})$ . In this paper, we generalize these results by considering the following class of Boolean functions over $\mathbb {F}_{2^{n}}$:

$$\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i} x^{r(2^{m}-1)+\frac{2^{n}-1}{3}i}) +T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), $$

where $n=2m$, m is odd, $b\in \mathbb {F}_{4}$, and $a_{r,i}\in \mathbb {F}_{2^{n}}$. With the restriction of $a_{r,i}\in \mathbb {F}_{2^{m}}$, we present a characterization of hyper-bentness of these functions in terms of crucial exponential sums. For some special cases, we provide explicit characterizations for some hyper-bent functions in terms of Kloosterman sums and cubic sums. Finally, we explain how our results on binomial, trinomial and quadrinomial hyper-bent functions can be generalized to the general case where the coefficients $a_{r,i}$ belong to the whole field $\mathbb {F}_{2^{n}}$.